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Example 1: MaxCut Input: graph G = (V, E) Goal: partition V into two parts A & B such that edges(A, B) is maximized Can also be formulated as Maximize objective, where x i ’s are 0-1 variables A fundamental (and very easily stated) combinatorial optimization problem G=(V,E) A B=V-A number of edges between A & B 3

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Tighten the relaxations [Khot-Vishnoi’05] Triangle ineq.’s do not improve the worst- case integrality gap for MaxCut In many occasions, triangle ineq.’s do help Famous example of SparsestCut on an n -vertex graph – IG of BasicSDP: – IG after triangle ineq.’s: [Arora-Rao-Vazirani’04] Can add even more constraints, leading to even better approximation guarantee 15

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Limitations of hierarchies We will prove theorems in the following style Fix a problem (e.g. MaxCut ), even using many levels (e.g. >100, >log n, >.1n ) of the hierarchy, the integrality gap is still bad – Design a ( MaxCut ) instance I – Prove real MaxCut of I small – Prove relaxation thinks MaxCut of I large I.e. the hierarchy does not give good approximation 26 True Optimum : 2 Relaxation Optimum : 9/4 ≈.889 Integrality gap (IG) = want it far from 1

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Motivation The big open problem in approximation algorithms research – Is it NP-hard to beat.878-approximation for MaxCut (Goemans- Williamson SDP)? – I.e. is Goemans-Williamson SDP optimal? 27

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Why is this interesting? 39 The big open question: Is Goemans-Williamson the best polynomial-time algorithm for MaxCut ? Evidence for Yes [KV’05, RS’09, BGHMRS’12] GW is optimal in Sherali-Adams+ hierarchy Evidence for No (our results) Hard instances from the left are solved by Lasserre-Parrilo

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The connection from algebraic proof complexity We relate power of Lasserre-Parrilo to power of an algebraic proof system – Sum-of-Squares (SOS) proof system – Proof system where the only way to deduce inequality is by p(x) 2 ≥ 0 – Dates back to Hilbert’s 17 th Problem 41 Given a multivariate polynomial that takes only non-negative values over reals, can it be represented as a sum of squares of rational functions?

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Algebraic proof systems – a new perspective for Lasserre-Parrilo Our method. Design a dual solution with small objective value What is Lasserre-Parrilo SDP? – Omitted due to time constraints… What is the dual SDP of Lasserre-Parrilo? Our key observation. (new view of the dual) SOS proof  dual solution i.e. SOS proof of MaxCut is small  dual value small Our goal. Translate the proof into SOS proof system Proofs of the known MaxCut IG [KV’05] Design a MaxCut instance I Prove real MaxCut of I small Prove relaxation thinks MaxCut of I large 43

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A comparison Construct integrality gaps Can use all mathematical proof techniques Give a deep proof to a deep theorem Our goal Can only use the limited axioms (as given by the SOS proof system) Give a “simple”(restricted) proof to a deep theorem What is the Sum-of-Squares (SOS) proof system? 44 Prove the MaxCut of the instance I is at most β

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Other works along this line [De-Mossel-Neeman’13] O(1) -level Lasserre-Parrilo almost exactly computes the optimum of the known MaxCut instances – Improves our work [O’Donnell-Zhou’13] which states that Lasserre-Parrilo gives better-than-.878 approximation [Barak-Brandão-Harrow-Kelner-Steurer-Zhou’12] O(1) -level Lasserre- Parrilo succeeds on all known UniqueGames instances [O’Donnell-Zhou’13] O(1) -level Lasserre-Parrilo succeeds on the known BalancedSeparator instances [Kauers-O’Donnell-Tan-Zhou’14] O(1) -level Lasserre-Parrilo succeeds on the hard instances for 3-Coloring Central problem in approximation algorithms A similar problem to SparsestCut 49

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Summary We utilize the connection between convex programming relaxations and theory of algebraic proof complexity – Lasserre-Parrilo solves the hardest known instances for MaxCut, UniqueGames, BalancedSeparator, 3-Coloring, … – Motivates study of SOS proof system to further understand power of Lasserre-Parrilo – Optimality of BasicSDP ( Goemans-Williamson ) seems more mysterious 50

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Future directions Maybe No? – Lasserre-Parrilo better approximation for all MaxCut instances? – We made initial step towards this direction Maybe Yes? – We gave insight in designing integrality gap instances: avoid the power of SOS proof system! 51 The big open question: Is Goemans-Williamson the best polynomial-time algorithm for MaxCut ? Our first step: Is Goemans-Williamson the best in Lasserre-Parrilo hierarchy?